Georgi D Raikov

Pontificia Universidad Católica de Chile | UC · Departamento de Matemática

We consider the self-adjoint Landau Hamiltonian H_0 in L^2(\mathbb{R}) whose spectrum consists of infinitely degenerate eigenvalues \Lambda_q , q \in \mathbb{Z}_+ , and the perturbed Landau Hamiltonian H_\upsilon = H_0 + \upsilon\delta_\Gamma , where \Gamma \subset \mathbb{R} is a regular Jordan C^{1,1} -curve and \upsilon \in L^p(\Gamma;\mathbb R)...

In this article, we provide the spectral analysis of a Dirac-type operator on Z2 by describing the behavior of the spectral shift function associated with a sign–definite trace–class perturbation by a multiplication operator. We prove that it remains bounded outside a single threshold and obtain its main asymptotic term in the unbounded case. Inter...

In this article, we provide the spectral analysis of a Dirac-type operator on $\mathbb{Z}^2$ by describing the behavior of the spectral shift function associated with a sign-definite trace-class perturbation by a multiplication operator. We prove that it remains bounded outside a single threshold and obtain its main asymptotic term in the unbounded...

We consider the self-adjoint Landau Hamiltonian $H_0$ in $L^2(\mathbb{R}^2)$ whose spectrum consists of infinitely degenerate eigenvalues $\Lambda_q$, $q \in \mathbb{Z}_+$, and the perturbed operator $H_\upsilon = H_0 + \upsilon\delta_\Gamma$, where $\Gamma \subset \mathbb{R}^2$ is a regular Jordan $C^{1,1}$-curve, and $\upsilon \in L^p(\Gamma;\mat...

This proceedings volume contains peer-reviewed, selected papers and surveys presented at the conference Spectral Theory and Mathematical Physics (STMP) 2018 which was held in Santiago, Chile, at the Pontifical Catholic University of Chile in December 2018. The original works gathered in this volume reveal the state of the art in the area and reflec...

We consider the Schrödinger operator \(H_0\) with constant magnetic field B of scalar intensity \(b>0\), self-adjoint in \(L^2({{\mathbb {R}}}^3)\), and its perturbations \(H_+\) (resp., \(H_-\)) obtained by imposing Dirichlet (resp., Neumann) conditions on the boundary of the bounded domain \(\Omega _{\mathrm{in}} \subset {{\mathbb {R}}}^3\). We i...

We consider the Landau Hamiltonian H 0 , self-adjoint in L 2 ( R 2 ), whose spectrum consists of an arithmetic progression of infinitely degenerate positive eigenvalues Λ q , q ∈ Z + . We perturb H 0 by a non-local potential written as a bounded pseudo-differential operator Op w ( V ) with real-valued Weyl symbol V, such that Op w ( V ) H 0 − 1 is...

We consider the 3D Schr\"odinger operator $H_0$ with constant magnetic field $B$ of scalar intensity $b>0$, and its perturbations $H_+$ (resp., $H_-$) obtained by imposing Dirichlet (resp., Neumann) conditions on the boundary of the bounded domain $\Omega_{\rm in} \subset {\mathbb R}^3$. We introduce the Krein spectral shift functions $\xi(E;H_\pm,...

We consider the Landau Hamiltonian $H_0$, self-adjoint in $L^2({\mathbb R}^2)$, whose spectrum consists of an arithmetic progression of infinitely degenerate positive eigenvalues $\Lambda_q$, $q \in {\mathbb Z}_+$. We perturb $H_0$ by a non-local potential written as a bounded pseudo-differential operator ${{\rm Op}^{\rm w}}(V)$ with real-valued We...

We consider the Dirichlet Laplacian (Formula presented.) on a 3D twisted waveguide with random Anderson-type twisting (Formula presented.). We introduce the integrated density of states (Formula presented.) for the operator (Formula presented.), and investigate the Lifshits tails of (Formula presented.), i.e. the asymptotic behavior of (Formula pre...

We consider a 2D Pauli operator with almost periodic field $b$ and electric potential $V$. First, we study the ergodic properties of $H$ and show, in particular, that its discrete spectrum is empty if there exists a magnetic potential which generates the magnetic field $b - b_{0}$, $b_{0}$ being the mean value of $b$. Next, we assume that $V = 0$,...

We consider the Krein Laplacian on a regular bounded domain Ω ⊂ ℝd, d ≥ 2, perturbed by a real-valued multiplier V vanishing on the boundary. Assuming that V has a definite sign, we investigate the asymptotics of the functions counting the eigenvalues of K +V which converge to the origin from below or from above. We show that the effective Hamilton...

I will consider the 3D Schrödinger operator with constant magnetic field, perturbed by a rapidly decaying electric potential. First, I will discuss the asymptotic behavior of the corresponding Krein spectral shift function (SSF) near the Landau levels which play the role of spectral thresholds. I will show that the SSF has singularities near these...

We consider Schrödinger operators with a random potential which is the square of an alloy-type potential. We investigate their integrated density of states and prove Lifshits tails. Our interest in this type of models is triggered by an investigation of randomly twisted waveguides.

We consider the Krein Laplacian on a regular bounded domain Ω ⊂ R d , d ≥ 2, perturbed by a real-valued multiplier V vanishing on the boundary. Assuming that V has a definite sign, we investigate the asymptotics of the functions counting the eigenvalues of K +V which converge to the origin from below or from above. We show that the effective Hamilt...

We consider harmonic Toeplitz operators $T_V = PV:{\mathcal H}(\Omega) \to {\mathcal H}(\Omega)$ where $P: L^2(\Omega) \to {\mathcal H}(\Omega)$ is the orthogonal projection onto ${\mathcal H}(\Omega) = \left\{u \in L^2(\Omega)\,|\,\Delta u = 0 \; \mbox{in}\;\Omega\right\}$, $\Omega \subset {\mathbb R}^d$, $d \geq 2$, is a bounded domain with $\par...

Abstract. The main goal of the course is to show the important role of the Berezin-Toeplitz operators in the spectral and scattering theory of magnetic quantum Hamiltonians. First, I will consider the magnetic Schrödinger operator H, and will recall some of its basic properties such as the gauge invariance and the diamagnetic inequality. I will all...

We consider Schr\"{o}dinger operators on $L^{2}({\mathbb R}^{d})\otimes L^{2}({\mathbb R}^{\ell})$ of the form $ H_{\omega}~=~H_{\perp}\otimes I_{\parallel} + I_{\perp} \otimes {H_\parallel} + V_{\omega}$, where $H_{\perp}$ and $H_{\parallel}$ are Schr\"{o}dinger operators on $L^{2}({\mathbb R}^{d})$ and $L^{2}({\mathbb R}^{\ell})$ respectively, an...

The present volume contains the Proceedings of the International Conference on Spectral Theory and Mathematical Physics held in Santiago de Chile in November 2014. Main topics are: Ergodic Quantum Hamiltonians, Magnetic Schrödinger Operators, Quantum Field Theory, Quantum Integrable Systems, Scattering Theory, Semiclassical and Microlocal Analysis,...

Preprint arXiv:1501.06865, v.3

We consider the Schr\"odinger operator $H_{\eta W} = -\Delta + \eta W$, self-adjoint in $L^2({\mathbb R}^d)$, $d \geq 1$. Here $\eta$ is a non constant almost periodic function, while $W$ decays slowly and regularly at infinity. We study the asymptotic behaviour of the discrete spectrum of $H_{\eta W}$ near the origin, and due to the irregular deca...

We consider metric perturbations of the Landau Hamiltonian. We investigate the asymptotic behaviour of the discrete spectrum of the perturbed operator near the Landau levels, for perturbations with power-like decay, exponential decay or compact support.

This note is based on the results contained in [1] and [2]. We consider the Landau Hamiltonian perturbed by an electric potential V that decays at infinity. The spectrum of the perturbed operator consists of eigenvalue clusters which accumulate to the Landau levels. We distinguish two regimes: a short-range and a long-range one, and estimate the ra...

We consider the twisted waveguide $\Omega_\theta$, i.e. the domain obtained by the rotation of the bounded cross section $\omega \subset {\mathbb R}^{2}$ of the straight tube $\Omega : = \omega \times {\mathbb R}$ at angle $\theta$ which depends on the variable along the axis of $\Omega$. We study the spectral properties of the Dirichlet Laplacian...

In this survey article we consider the operator pair $(H,H_0)$ where $H_0$ is the shifted 3D Schr\"odinger operator with constant magnetic field, $H : = H_0 + V$, and $V$ is a short-range electric potential of a fixed sign. We describe the asymptotic behavior of the Krein spectral shift function (SSF) $\xi(E; H,H_0)$ as the energy $E$ approaches th...

We consider a twisted quantum waveguide, i.e. a domain of the form Omega(theta) := r(theta)omega x R where omega subset of R-2 is a bounded domain, and r(theta) = r(theta) (x(3)) is a rotation by the angle theta (x(3)) depending on the longitudinal variable x(3). We investigate the nature of the essential spectrum of the Dirichlet Laplacian H-theta...

We consider the Landau Hamiltonian perturbed by an electric potential V which decays sufficiently fast at infinity. The spectrum of the perturbed Hamiltonian consists of clusters of eigenvalues which accumulate to the Landau levels. Applying a suitable version of the anti-Wick quantization, we investigate the asymptotic distribution of the eigenval...

We consider the Landau Hamiltonian perturbed by a long-range electric potential $V$. The spectrum of the perturbed operator consists of eigenvalue clusters which accumulate to the Landau levels. First, we obtain an estimate of the rate of the shrinking of these clusters to the Landau levels as the number of the cluster $q$ tends to infinity. Furthe...

Let $H_{0, D}$ (resp., $H_{0,N}$) be the Schroedinger operator in constant magnetic field on the half-plane with Dirichlet (resp., Neumann) boundary conditions, and let $H_\ell : = H_{0, \ell} - V$, $\ell =D,N$, where the scalar potential $V$ is non negative, bounded, does not vanish identically, and decays at infinity. We compare the distribution...

We consider the Landau Hamiltonian (i.e. the 2D Schroedinger operator with constant magnetic field) perturbed by an electric potential V which decays sufficiently fast at infinity. The spectrum of the perturbed Hamiltonian consists of clusters of eigenvalues which accumulate to the Landau levels. Applying a suitable version of the anti-Wick quantiz...

We consider the meromorphic operator-valued function 1-K(z) = 1-A(z)/z where A(z) is holomorphic on the domain D, and has values in the class of compact operators acting in a given Hilbert space. Under the assumption that A(0) is a selfadjoint operator which can be of infinite rank, we study the distribution near the origin of the characteristic va...

We consider a twisted quantum waveguide i.e. a domain of the form \Omega_{\theta} : = r_\theta \omega \times R, where \omega \subset R^2 is a bounded domain, and r_\theta = r_\theta(x_3) is a rotation by the angle \theta(x_3) depending on the longitudinal variable x_3. We investigate the nature of the essential spectrum of the Dirichlet Laplacian H...

We investigate the edge conductance of particles submitted to an Iwatsuka magnetic field, playing the role of a purely magnetic barrier. We also consider magnetic guides generated by generalized Iwatsuka potentials. In both cases, we prove quantization of the edge conductance. Next, we consider magnetic perturbations of such magnetic barriers or gu...

We consider the unperturbed operator $H_0: = (-i \nabla - {\bf A})^2 + W$, self-adjoint in $L^2({\mathbb R}^2)$. Here ${\bf A}$ is a magnetic potential which generates a constant magnetic field $b>0$, and the edge potential $W = \bar{W}$ is a ${\mathcal T}$-periodic non constant bounded function depending only on the first coordinate $x \in {\mathb...

We consider a 2D Schrödinger operator H_0 with constant magnetic field defined on a strip of finite width. The spectrum of H_0 is absolutely continuous and contains a discrete set of thresholds. We perturb H_0 by an electric potential V, and establish a Mourre estimate for H = H_0 + V when V is periodic in the infinite direction of the strip, or de...

We consider the unperturbed operator $H_0 : = (-i \nabla - A)^2 + W$, self-adjoint in $L^2(\R^2)$. Here $A$ is a magnetic potential which generates a constant magnetic field $b>0$, and the edge potential $W$ is a non-decreasing non constant bounded function depending only on the first coordinate $x \in \R$ of $(x,y) \in \R^2$. Then the spectrum of...

In this note we consider a Landau Hamiltonian perturbed by a random magnetic potential of Anderson type. For a given number of bands, we prove the existence of both strongly localized states at the edges of the spectrum and dynamical delocalization near the center of the bands in the sense that wave packets travel at least at a given minimum speed....

The aim of this note is to review recent articles on the spectral properties of magnetic Schrödinger operators. We consider H0, a 3D Schrödinger operator with constant magnetic field, an H0, a perturbation of H0 by an electric potential which depends only on the variable along the magnetic field. Let H (resp H) be a short range perturbation of H0 (...

We consider a twisted quantum wave guide i.e., a domain of the form Ωθ: = rθ ω × ℝ where ω ⊂ ℝ2 is a bounded domain, and rθ = rθ(x3) is a rotation by the angle θ(x3) depending on the longitudinal variable x3. We are interested in the spectral analysis of the Dirichlet Laplacian H acting in L2(Ωθ). We suppose that the derivative θ̇ of the rotation a...

We consider the 3D Pauli operator with nonconstant magnetic field B of constant direction, perturbed by a symmetric matrix-valued electric potential V whose coefficients decay fast enough at infinity. We investigate the low-energy asymptotics of the corresponding spectral shift function. As a corollary, for generic negative V, we obtain a generaliz...

We show that the Landau levels cease to be eigenvalues if we perturb the 2D Schrödinger operator with a constant magnetic field, by bounded electric potentials of fixed sign. We also show that, if the perturbation is not of fixed sign, then any Landau level may be an eigenvalue of the perturbed problem.

We consider a class of translationally invariant 3D Pauli operators G with magnetic fields with circular integral curves. We show that the spectrum of G is purely absolutely continuous, coincides with (0,1), and has an infinite multiplicity.

We consider a 2D Schrödinger operator H 0 with constant magnetic field, on a strip of finite width. The spectrum of H 0 is absolutely continuous, and contains a discrete set of thresholds. We perturb H 0 by an electric potential V which decays in a suitable sense at infinity, and study the spectral properties of the perturbed operator H=H 0 +V. Fir...

We consider a 3D magnetic Schrödinger operator having infinitely many eigenvalues of infinite multiplicity, embedded in the continuous spectrum. We perturb this operator by a relatively compact potential and analyse the transition of these eigenvalues into a “cloud” of resonances. Several differential approaches are employed. First we consider reso...

We consider the Hamiltonian $H$ of a 3D spinless non-relativistic quantum particle subject to parallel constant magnetic and non-constant electric field. The operator $H$ has infinitely many eigenvalues of infinite multiplicity embedded in its continuous spectrum. We perturb $H$ by appropriate scalar potentials $V$ and investigate the transformatio...

We consider a 2D Schroedinger operator H0 with constant magnetic field, on a strip of finite width. The spectrum of H0 is absolutely continuous, and contains a discrete set of thresholds. We perturb H0 by an electric potential V which decays in a suitable sense at infinity, and study the spectral properties of the perturbed operator H = H0 + V . Fi...

In this survey article based on the papers [7, 10], and [8], we consider the 3D Schröxzdinger operator with constant magnetic field of intensity b > 0, perturbed by an electric potential V which decays fast enough at infinity, and discuss various asymptotic properties of the corresponding spectral shift function.

We consider three-dimensional Schrödinger operators with constant magnetic fields and ergodic electric potentials. We study the strong magnetic field asymptotic behaviour of the integrated density of states, distinguishing between the asymptotics far from the Landau levels, and the asymptotics near a given Landau level.

We consider the 3D Schr\"odinger operator $H = H_0 + V$ where $H_0 = (-i\nabla - A)^2$, $A$ is a magnetic potential generating a constant magnetic field of strength $b>0$, and $V$ is a short-range electric potential which decays superexponentially with respect to the variable along the magnetic field. We show that the resolvent of $H$ admits a mero...

We consider the 2D Landau Hamiltonian $H$ perturbed by a random alloy-type potential, and investigate the Lifshitz tails, i.e. the asymptotic behavior of the corresponding integrated density of states (IDS) near the edges in the spectrum of $H$. If a given edge coincides with a Landau level, we obtain different asymptotic formulae for power-like, e...

We consider the three-dimensional Schrodinger operator with constant magnetic field, perturbed by an appropriate short-range electric potential, and investigate various asymptotic properties of the corresponding spectral shift function (SSF). First, we analyse the singularities of the SSF at the Landau levels. Further, we study the strong magnetic...

We consider the three-dimensional Schrödinger operators H0 and H where H0=(i∇+A)2-b, A is a magnetic potential generating a constant magnetic field of strength b>0, and H=H0+V where V∈L1(R3R) satisfies certain regularity conditions. Then the spectral shift function xi(EH,H0) for the pair of operators H, H0 is well-defined for energies E!=2qb, q∈Z+....

We consider the three-dimensional Schrödinger operators $$ H_0 $$ and $$ H_\pm $$ where $$ H_{0} = (i\nabla + A)^{2} - b $$ , A is a magnetic potential generating a constant magnetic field of strength $$ b > 0 $$ , and $$ H_{\pm} = H_{0} \pm V $$ where $$ b \geq 0 $$ decays fast enough at infinity. Then, A. Pushnitski’s representation of the spectr...

We prove that the integrated density of states (IDS) for the randomly perturbed Landau Hamiltonian is Hölder continuous at all energies with any Hölder exponent < q < 1. The random Anderson-type potential is constructed with a non-negative, compactly supported single-site potential u. The distribution of the iid random variables is required to be a...

The three-dimensional Schrodinger operator H with constant magnetic field of strength b> 0 is considered under the assumption that the electric potential V ∈ L1(R3) admits certain power-like estimates at infinity. The asymptotic behavior as b →∞ of the spectral shift function ξ(E; H, H0) is studied for the pair of operators (H, H0 )a t the energies...

We consider the Pauli operator H(b,V ) acting in L2 (R2;C2). We describe a class of oscillating magnetic fields b for which the ground state of the unperturbed operator H(b,0) which coincides with the origin, is an isolated eigenvalue of infinite multiplicity. Under the assumption that the matrix-valued electric potential V has a definite sign and...

We consider the Schrodinger operator H(V ) on L ) or L ) with constant magnetic field, and a class of electric potentials V which typically decay at infinity exponentially fast or have a compact support. We investigate the asymptotic behaviour of the discrete spectrum of H(V ) near the boundary points of its essential spectrum. If V decays like a G...

We consider the Schroedinger operator H on L^2(R^2) or L^2(R^3) with constant magnetic field and electric potential V which typically decays at infinity exponentially fast or has a compact support. We investigate the asymptotic behaviour of the discrete spectrum of H near the boundary points of its essential spectrum. If the decay of V is Gaussian...

. We consider the three-dimensional Schrodinger operator with strong constant magnetic field and random electric potential. We investigate the asymptotic behaviour of its integrated density of states near the qth Landau level, for any fixed q ? 1. 1 Introduction In this paper we consider the three-dimensional Schrodinger operator with constant magn...

this article is to present some of the authors' recent results on the asymptotic behaviour of the discrete spectrum of the Schroedinger, Pauli, and Dirac operators in strong magnetic fields perturbed by electric potentials which decay at infinity. Since the article is oriented towards readers who are not necessarily specialists in the field, we all...

We consider the three-dimensional Schrodinger operator with constant magnetic field and bounded random electric potential. We investigate the asymptotic behaviour of the integrated density of states for this operator as the norm of the magnetic field tends to infinity. On considere l'operateur de Schrodinger tridimensionnel avec un champ magnetiqu...

This article could be regarded as a supplement to [11] where we considered the Schr"odinger operator with constant magnetic field and decaying electric potential, and studied the asymptotic behaviour of the discrete spectrum as the coupling constant of the magnetic field tends to infinity. To describe this behaviour when the kernel of the magnetic...

We consider the three-dimensional Dirac operator H with constant magnetic field and electric potential which decays at infinity. We study the asymptotic behaviour of the discrete spectrum of H as the norm of the magnetic field grows unboundedly.

We consider the Pauli operator H(alpha) := (Sigma(j)(m) = 1 sigma(j) (- i partial derivative/partial derivative x(j) -mu A(j)))(2) + V selfadjoint in L-2(R-m; C-2), m = 2, 3. Here sigma(j), j = 1,..., m, are the Pauli matrices, A := (A(1),..., A(m)) is the magnetic potential, mu > 0 is the coupling constant, and V is the electric potential which de...

We consider the two-dimensional Schrödinger operator with generically bounded magnetic field, perturbed by a scalar potential -gV, g ≥ 0 being the coupling constant. We assume that V is non-negative and decays rapidly at infinity (e.g., V(x) (equivalent to) |x|-α, α > 2, as |x| → ∞). We examine the asymptotic behaviour as g → ∞ of the eigenvalues s...

The essential spectrum of the force operator of the ideal linear magnetohydrodynamics has been extensively studied in the mathematical literature. However, all the rigorous from mathematical point of view works on this topic concern the magnetohydrodynamic (MHD) model of a plasma confined in a bounded domain O{sub p} {contained_in} R{sup 3} with pe...

We consider the discrete spectrum of the Schrödinger operator ℌ h,μ :=(ih∇+μA) 2 -V where A is the magnetic potential, -V is the electric potential, h is the Planck constant, and μ is the magnetic-field coupling constant. We study the asymptotic behaviour of the number of the eigenvalues of ℌ h,μ smaller than λ≤0 as h↓0, μ>0 being fixed, or μ↓0, h>...

We consider the Schrdinger operator with constant full-rank magnetic field, perturbed by an electric potential which decays at infinity, and has a constant sign. We study the asymptotic behaviour for large values of the electric-field coupling constant of the eigenvalues situated in the gaps of the essential spectrum of the unperturbed operator.

We consider the Schrdinger operatorH=–+W+V acting inL 2( m ),m2, with periodic potentialW perturbed by a potentialV which decays slowly at infinity. We study the asymptotic behaviour of the discrete spectrum ofH near any given boundary point of the essential spectrum.

We consider in L 2 (ℝ m ) the Schrödinger operator H:=(i∇+A) 2 -V with constant magnetic field tensor B={∂ i A j -∂ j A i } i,j=1 m such that dimKerB>0, and electric potential V satisfying the asymptotics V(X)=g|X| -2 +O|X| -2-ε as |X|→∞, with g>0 and ε>0. We establish criteria which guarantee that the isolated eigenvalues of H do not accumulate to...

We consider the Schrödinger operator with electric potential V, which decays at infinity, and magnetic potential A. We study the asymptotic behaviour for large values of the electric field coupling constant of the eigenvalues situated under the essential-spectrum lower bound. We concentrate on the cases of rapidly decaying V (e.g. V ∈ L m/2(ℝm ) fo...

We consider an ideal linear magnetohydrodynamic model with translational symmetry and investigate the spectral properties of the force operator A k with a fixed longitudinal wavenumber k. We establish that the essential spectrum of A k consists of two bounded segments (slow magnetosonic and Alfvén continuum). Next we show that the isolated eigenval...

One considers the frequency spectrum of the linear oscillations of a perfectly conducting nonviscous plasma with a nonzero pressure. It is shown that the essential spectrum for each of the harmonic modes is not empty and can consist of at most two finite intervals (or points). The discrete spectrum contains an infinite number of eigenvalues, diverg...

We consider the discrete spectrum of the selfadjoint Schrdinger operatorA h =–h 2 +V defined inL 2(m) with potentialV which steadies at infinity, i.e.V(x)=g+|x|– f(1+o(1)) as |x| for>0 and some homogeneous functionsg andf of order zero. Let h (),0, be the total multiplicity of the eigenvalues ofA h smaller thanM–, M being the minimum value ofg...

The linear oscillations of an ideally conducting nonviscous plasma in an external direct magnetic field are examined using a magnetohydrodynamic model. In particular, attention is given to the frequency spectrum corresponding to a fixed harmonic mode. The asymptotic behavior of the discrete spectrum is investigated at infinity and near the material...

We consider the Hamiltonian H of a 3D spinless non-relativistic quantum particle subject to parallel constant magnetic and non-constant electric field. The operator H has infinitely many eigenvalues of infinite multiplicity embedded in its continuous spectrum. We perturb H by appropriate scalar potentials V and investigate the transformation of the...